Convergence of the Method of Piecewise Linear Approximations and Collocations for a Two-Dimensional Hypersingular Integral Equation on a Set With Boundary

Abstract

A hypersingular integral equation on a convex bounded set on the plane with an integral understood in the sense of a finite part in the sense of Hadamard is considered. Equations of this type, in particular, arise when solving the Neumann boundary value problem for the Lapalse and Helmholtz equations on a flat screen in the case where the solution is sought in the form of a double layer potential. To numerically solve the equation, a numerical scheme is used based on piecewise linear approximation of the unknown function on a triangular conformal mesh and the collocation method. The uniform convergence of numerical solutions to an exact solution on a grid when the maximum cell diameter tends to zero has been proven.